File:Bifilar 3-2 pattern in 3 -phase 13-8 knots 800x600.mp4

// Persistence of Vision Ray Tracer Scene Description File // File: Groupkos.dev.geomagic_3-2_pattern_in_bifilar_13-8_torus_knots_800x600.pov // POV-Ray Vers: 3.7 RC7 win64

// Desc: Polyphase 13:8 torus knot group w/ colored halves (bifilar). // Auth: DonEMitchell -- a electrified knot group fanatic (XenoEngineer) // Edited: March 9, 2020 Revisited and reorganized for posting at groupkos.com/dev/category:povray // Edited: April 21, 2020 --added cylinder knot loop elements. // Edited: April 22, 2020 for groupkos/dev propagation animation // Edited: July 23, 2022 cleaned up for dev wiki posting. // Date: November 29, 2023 --chiral grouping illustration

// -- HOW TO invoke an animation sequence --

// Put the control-code below into the POV-Ray text-box above the POV-Ray editor // (without the comment slashes). // Or from the menu: Render - Edit settings/Render. // Then click Run (Alt G). Search POV-Ray help for 'Animation'. // Or, right-click on the line below, and select "Copy [...] to the command line"

// +SF1 +EF299 +KFI1 +KFF299 +W800 +H600 +A

// The above POV-Ray control code causes POV-Ray to restart this file 299 times, // creating 299 image frames. // Each time the animation sequence runs, a variable named 'clock' is advanced // from zero to one (0-1). // The clock variable is used below to modulate parameters affecting the rendering. // Note: POV-Ray does not generate an animated GIF, which is done with external utility.

1. version 3.7;

1. include "colors.inc"

global_settings {

   max_trace_level 10 // Reflection levels before blackness.   
   assumed_gamma 1.2
   charset utf8
   // Used in global_settings, sets an overall brightness/ambient light level in the scene
   ambient_light color rgb <1,1,1> * .5
   } 

1. default{ finish{ ambient .5 diffuse 0.5 }}

sky_sphere{ pigment { gradient <-1,1,-1>

                     color_map { [0.00 rgb <0.6,0.7,0.80>*.2]
                                 [0.35 rgb <0.0,0.1,0.8>*.2]
                                 [0.65 rgb <0.0,0.1,0.8>*.2]
                                 [1.00 rgb <0.6,0.7,1.0>*.2] 
                               } 
                     scale 2         
                   } // end of pigment

}

//--------------------------- Macros ---------------------------------

// Axes macro: length and radii of axes, and boolean flags to include/exclude axes

1. macro axes(length, radii, X, Y, Z)

   #if (X) cylinder{ 0, < length,         0,       0>, radii texture { pigment { color rgb  < .75,-.1,-.1     > }}} #end //  X axis, red    
   #if (X) cylinder{ 0, <-length,         0,       0>, radii texture { pigment { color rgbt <   1,  0,  0, .6 > }}} #end // -X axis red translucent -adjust xmit                                                          
   #if (Y) cylinder{ 0, <      0,  length/2,       0>, radii texture { pigment { color rgb  < -.1, .75,-.1    > }}} #end //  Y axis, green    
   #if (Y) cylinder{ 0, <      0, -length/2,       0>, radii texture { pigment { color rgbt <   0, .8,  0, .6 > }}} #end //  Y axis, green translucent
   #if (Z) cylinder{ 0, <      0,         0,  length>, radii texture { pigment { color rgb  <   0,  0,  2     > }}} #end //  Z axis, Blue    
   #if (Z) cylinder{ 0, <      0,         0, -length>, radii texture { pigment { color rgbt <  .1, .1, .4, .6 > }}} #end // -Z axis blue translucent     

1. end //macro axes()

1. macro torusKnotBifilar(objectBifilarA, objectBifilarB, objLugA, objLugB, radiusConductor, radiusConnector, major, minor, poloid, toroid, polyphase, delta)

   union{
       // Render objectBifilarA and B as connector lugs
       object{objLugA translate x*(major+minor+radiusConnector*0)}
       // Render a objectBifilarB as a connector lug
       object{objLugB translate -x*(major+minor+radiusConnector*0)}
       
       // Draw both bifilar objects in one loop from zero through 180 degrees.
       #local  n  = 0;                             // n Is the while loop index.
       
       // The clock variable modulates completion length of while loop to animate propagation of bifilar halves.
       #while (n <= (180 * poloid * clock))        // Loop positive 180 degrees, and draw the objectBifilarB, below, looping the negative 180.
           #local phaz = n * toroid/poloid;        // Divide out poloid (multiplied already into n) to obtain toroidal rotation.
           
           // 1st Bifilar half as objectBifilarA
           #if( 01 )
               object{ objectBifilarA
                   // Move the objectBifilar away from the origin on the X axis by the minor radius.
                   translate  x*minor
                   // Toroidally rotate objectBifilar around Z axis in the XY plane.
                   rotate <0,0,phaz>
                   // Move the rotated objectBifilar the distance of the major radius in the X-axis.    
                   translate x*major
                   // Rotate the object about the center Y axis of the torus.                                     
                   rotate y*n
               }
            #end             
           #if( 01 )
               object{ objectBifilarB
                   // Move the objectBifilar away from the origin on the X axis by the minor radius.
                   translate  x*minor
                   // Toroidally rotate objectBifilar around Z axis in the XY plane.
                   rotate <0,0,phaz>
                   // Move the rotated objectBifilar the distance of the major radius in the X-axis.    
                   translate x*major
                   // Rotate the object about the center Y axis of the torus.                                     
                   rotate y*n
                   rotate y*180
               }
            #end             
            
           // Counter-toroid
           #if( 0 )
               object{ objectBifilarA
                   // Move the objectBifilar away from the origin on the X axis by the minor radius.
                   translate   x*minor
                   // Toroidally rotate objectBifilar around Z axis in the XY plane.
                   rotate <0,0,-phaz>
                   // Move the rotated objectBifilar the distance of the major radius in the X-axis.    
                   translate x*major           
                   // Rotate the object about the center Y axis of the torus.                                     
                   rotate y*n
               }
            #end
           // 2nd Bifilar half as objectBifilarB (counter-toroid and counter-poloid rotations)
            #if( 0 )
               object{ objectBifilarB
                   // Move the objectBifilar away from the origin on the X-axis by the minor radius.
                   translate   x*minor
                   // Toroidally Counter-rotate objectBifilarB around Z-axis in the XY-plane.
                   rotate <0,0,-phaz>
                   // Move the rotated objectBifilar the distance of the major radius in the X-axis.    
                   translate x*major
                   // Counter-rotate the objectBifilarB about the center Y-axis of the torus.                                     
                   rotate -y*n
                   rotate -y*(180)                                        
               }                
           #end                
         
           #if(showQuadrantConnectors)
                sphere{
                   0,radiusConnector
                   translate  x*(radiusMinor + radiusMajor)
                   rotate y*90
                   texture { pigment{rgbt<1,1,0,.0975>} } 
                }
                sphere{
                   0,radiusConnector
                   translate  x*(radiusMinor + radiusMajor)
                   rotate -y*90
                   texture { pigment{rgbt<1,1,0,.0975>} } 
                   }
           #end
                       
           #local n = n + delta;
       #end // while
       
       rotate y*polyphase // Rotate the completed knot to its phase-angle about the torus center Y-axis.
   } // end union

1. end // macro torusKnotBifilar()

// ----------------- Optics --------------------

camera {

   //orthographic
   //location    <2*27-27*clock,8,0>
   location    z*28
   look_at     y*0
   angle       58
   sky         y
   right       x*image_width/image_height

}

// general light definition light_source {

 <15, 10, 10>          // position of the light source
 color rgb 1.0*3      // color of the light
 shadowless

}

// ---------------- Scene ------------------------

// 13/8 = 1.625 // Phi = 1.6180339887498948482045868343656 // Difference = 1.625 - Phi // Difference = 0.00696601125010515179541316563436

// Torus knot parameters (loop/twist) //

1. declare poloid = 13;
2. declare toroid = 8;

//#declare poloid = 12 + 1*clock; // clock sequences from 0 to 1 //#declare toroid = 8;

1. declare phaseCount = 3;

// Delta as increment between objects tracing torus loop in the torus knot macro. // The knot is a string of overlapping spheres. Delta sets the separation.

1. declare plotDelta = .05;

// Torus dimensions // Torus parameters and knot ratio (winding number) construct a 'golden orthogonal (ish) Fibonacci torus knot' (GOFTK) // FTK: Neighboring Fibonacci-sequence numbers (1,1,2,3,5,8,13,21...) // Poloidal rotations = Fn+1 // Toroidal rotations = Fn // Torus knot ratio: (Fn+1)/(Fn) = golden ratio (almost) --The knot ratio approximates Phi = 13/8 = 1.6025

1. declare Phi = pow(5,.5) * .5 + .5; // Upper case Phi = 1.618...
2. declare PhiIndex = 4; // Set the radix of Phi for the major radius.

// Calculate orthogonal major/minor radii.

1. declare radiusMajor = pow(Phi, PhiIndex );
2. declare radiusMinor = radiusMajor-pow(Phi, PhiIndex - 4 ); // Four degrees of Phi smaller.
3. declare ySquash = 1;

1. declare radiusConnector = .45;
2. declare radiusConductor = .15;
3. declare showConnectors = 0;
4. declare showQuadrantConnectors = 0;

// Toggle parts of the scene.

1. declare showBaseTorus = 10;
2. declare showTorusKnots = 01;
3. declare showAxes = 0;
4. declare axesRadius = 0.1;

1. declare colorA = Green*.5;
2. declare colorB = Orange*.5;

// Center objects on the origin for passing to the macro

1. declare objA = sphere{0, radiusConductor texture{pigment{color colorA}}};
2. declare objB = sphere{0, radiusConductor texture{pigment{color colorB}}};
3. declare objLugA = sphere{ 0, radiusConnector texture{pigment{color colorA }} }
4. declare objLugB = sphere{ 0, radiusConnector texture{pigment{color colorB }} }

union{

   // Bifilar torus knots.    
   #if(showTorusKnots) 
       #local phaseAngle = 360/phaseCount;
       
       #local j = 0;   // While-loop index
       // Render multiple bifilar phases rotated apart by multiples of phaseAngle //
       union{
           #while(j < phaseCount)
            // Call torus knot macro 
            // torusKnotBifilar ( objA, objB, objLugA, objLugB, radiusConductor, radiusConnector, major,       minor,       poloid, toroid, polyphase,    delta)
               torusKnotBifilar ( objA, objB, objLugA, objLugB, radiusConductor, radiusConnector, radiusMajor, radiusMinor, poloid, toroid, j*phaseAngle, plotDelta)                
             //object{torusKnotBifilar ( objA, objB, objLugA, objLugB, radiusConductor, radiusConnector, radiusMajor, radiusMinor, poloid, toroid, j*phaseAngle, plotDelta) rotate -y*90}
               #declare j = j + 1; // Increment loop index
           #end 
           scale y*ySquash 
           //clipped_by{plane{-x,0}}
       }
   #end // Moebius bifilar torus knots.    
  
   #if(showBaseTorus)
       torus{ radiusMajor,radiusMinor *.9994
              texture { pigment{rgbt<1,1,0,.985>} } 
              //texture { pigment{rgbt<1,1,1,.07> } }   
              //clipped_by{plane{z,0}}
              scale y*ySquash
       }
   #end    
   
   #if(showAxes)
       axes(radiusMajor+radiusMinor*1.0, axesRadius,1,1,1)
   #end
   

} // end scene union{}